MATH 151 TEST 3 12 13 07 FALL 2007 Remember to keep your work neat and orderly Show all of your work NO WORK NO CREDIT Read each question carefully and be sure to answer the question that was asked Calculator answers are not acceptable Name f x sin 2 x rd 1 Find the 3 degree Taylor polynomial for the function other known series to do this Note sin 4 cos 4 1 pts a 8 at Do not use any 2 9 2 Evaluate each of the following convergent series note the first series starts at n 1 5 a 3n n 1 b 1 1 1 1 1 3 4 5 6 2 2 2 2 2 2 6 6 3 Do the following series converge conditionally converge absolutely or diverge Justify a 7 3 n 1 n 1 n n 3 2n n 1 c 1 n 1 b n n 1 4 5 6 4 Without using another Taylor series find the infinite Taylor series for the function at a 0 Show the terms through x 3 and show the general form of the nth term 3 x 4 n n n 1 5 Find the interval of convergence for the power series 6 f x e 3 x 9 9 u 3 1 v 2 4 a Sketch u and v in standard position both starting at the origin Sketch the vector represented by u v using the parallelogram rule 5 b Find a unit vector in the opposite direction of u 3 c Find u v 2 d Find a non zero vector perpendicular to u 3 7 If u 2 1 3 v 1 2 1 find each of the following You may and should use any results from a previous part from this question to help with the current part a u v b u v c The angle between u and v to the nearest hundredth of a radian 5 d Two different unit vectors that are perpendicular to both u and v 4 e Write u as the sum of two vectors one that is parallel to v and one that is perpendicular to v 8 If 2 5 a 2 1 2 b 3 1 4 c 5 3 1 find the triple scalar product a b c 7 6 9 Give an example if possible of a sequence a n that converges to zero but the series an n 1 diverges 4